There are a whole bunch of related types of football. Most of them allow handling of the ball. And they are all quite properly referred to as varieties of "football".

The one which is most popular in any country actually tends to be called Football. In most countries that is the game historically called Association Football, Soccer for short, which does outlaw handling the ball, except by the goalkeeper.

People in England have mostly tended to speak indifferently of either Soccer or Football. Soccer more especially when they want to avoid ambiguity, and confusion with Rugby Football.

Presumably as Soccer becomes the main game in the USA - (and I believe it is already actually played by far more people there than American Football), the same process will happen there. The teasing joke about the name is wearing a bit thin.